Is the deformation limit of Ricci-flat Kahler manifolds Kahler?

Question

Let $X$ be a compact complex Kahler manifold with first real Chern class $c_1 = 0$. Consider a family $\pi : \mathcal X \to \Delta$ over the unit disc in $\mathbb C$, where the fibers $X_s$ are compact Kahler with $c_1(X_s) = 0$ for $s \not= 0$. Do we know that the central fiber $X_0$ is Kahler with $c_1 = 0$?

Some remarks:

a) The condition on the Chern class is topological, so the question is really if the central fiber is Kahler.

b) This is true for complex tori and K3 surfaces, though for K3 surfaces being Kahler is a consequence of the topological condition of having even first Betti number.

c) Kuranishi constructed an example of non-Kahler deformations of projective manifolds, but the manifolds in question were not Kahler-Einstein so that example does not apply here.

d) There exist non-Kahler compact complex manifolds with $c_1 = 0$, like the Iwasawa manifold. The Iwasawa manifold does not have the right first Betti number to provide a counterexample to the question. However, I've heard physicists have found many examples of non-Kahler manifolds of Calabi-Yau type, and maybe one of those does at least not have topological obstructions to being a counterexample?

[edit] Two more remarks:

e) For the special case of Calabi-Yau manifolds, Popovici gives that the central fiber is Moishezon. One could hope that a Moishezon manifold with the Hodge numbers of a Calabi-Yau manifold is Kahler, but this is false by an example of Oguiso. His example is however not homeomorphic to a projective manifold, leaving the question open.

f) One could try looking at Calabi-Yau threefolds, of which many examples are apparently known (I only know of complete intersections of appropriate degree in projective space). The case of a Calabi-Yau hypersurface in $\mathbb P^4$ is uninteresting, as they are rigid, so the central fiber is isomorphic to the general fiber.

Answer 1

There are counterexamples: a Moishezon manifold, which has trivial canonical class and is birationally equivalent to a hyperkahler manifold, is also deformationally equivalent to a hyperkaehler manifold (this is a result of Huybrechts, I am not sure if he stated it in this generality, but his proof certainly works). Such Moishezon manifolds can be non-Kaehler (see e.g. Periods of Enriques Manifolds, Keiji Oguiso, Stefan Schroeer, http://arxiv.org/abs/1010.0820, Proposition 6.2).

Answer 2

Let $\pi:X\to Y$ be a flat projective family of n-dimensional varieties with central fibre $X_0$ be Calabi-Yau variety with canonical singularities then all general fibres $X_y$ are Calabi-Yau varieties with at worst canonical singularites. See Fibration when central fibre is a Calabi-Yau variety with canonical singularities

The inverse of this statement also holds true with some additional assumption.

Let $\pi:(X,D)\to Y$ be a flat projective family of n-dimensional varieties with $K_{X/Y}+D\cong \mathcal O_X(D)$ (i.e. fibres are log Calabi-Yau varieties) and $(X,(X_0,D_0))$ be divisorially log terminal then the log Calabi-Yau central fibre $(X_0,D_0)$ has at worst canonical singularities if and only if there exists an irreducible component $N$ of $X_0$ with a resolution of singularities $\tilde N\to N$ with $H^0(\tilde N,K_{\tilde N})\neq 0$. In fact, if $f:(\tilde X_0,\tilde D_0)\to (X_0,D_0)$ is a resolution of singularities, then we have

$$K_{\tilde X_0}+\tilde D_0\thicksim \sum_{E}a_EE$$ with $a_E\geq 0$, it follow that $K_{\tilde X_0}+\tilde D_0$ is effective. Now it is not very hard by the Idea of Wang to show that the converse is also holds true and we just need to know $K_0+D_0$ is irreducible and trivial.

By the results of Shigeharu Takayama, Wang and V.Tosatti if $0$ lies at finite Weil-Petersson distance then central fibre $X_0$ is Calabi-Yau variety with canonical singularities.

Note that for polarized Calabi-Yau degeneration $f:X\to \mathbb D$, the central fobre $X_0$ is Calabi-Yau variety with canonical singularities if and only if the Tian's Kahler potential of Weil-Petersson metric be upper bounded.

$$\omega_{WP}=-\sqrt[]{-1}\partial_y\bar\partial_y\log\int_{X_y}|\Omega_y|^2$$